X-RAYS

Spacetime Intervals

2.5 X-RAYS

They consist of high-energy photons 2.6 X-RAY DIFFRACTION

How x-ray wavelengths can be determined 2.7 COMPTON EFFECT

Further confirmation of the photon model 2.8 PAIR PRODUCTION

Energy into matter 2.9 PHOTONS AND GRAVITY

Although they lack rest mass, photons behave as though they have gravitational mass

I

n our everyday experience there is nothing mysterious or ambiguous about the concepts of particle and wave.A stone dropped into a lake and the ripples that spread out from its point of impact apparently have in common only the ability to carry energy and momentum from one place to another. Classical physics, which mirrors the “physical reality” of our sense impressions, treats particles and waves as separate components of that reality. The mechanics of particles and the optics of waves are traditionally independent disciplines, each with its own chain of experiments and principles based on their results.

The physical reality we perceive has its roots in the microscopic world of atoms and molecules, electrons and nuclei, but in this world there are neither particles nor waves in our sense of these terms. We regard electrons as particles because they possess charge and mass and behave according to the laws of particle mechanics in such familiar de- vices as television picture tubes. We shall see, however, that it is just as correct to in- terpret a moving electron as a wave manifestation as it is to interpret it as a particle manifestation. We regard electromagnetic waves as waves because under suitable cir- cumstances they exhibit diffraction, interference, and polarization. Similarly, we shall see that under other circumstances electromagnetic waves behave as though they con- sist of streams of particles. Together with special relativity, the wave-particle duality is central to an understanding of modern physics, and in this book there are few argu- ments that do not draw upon either or both of these fundamental ideas.

2.1 ELECTROMAGNETIC WAVES

Coupled electric and magnetic oscillations that move with the speed of light and exhibit typical wave behavior

In 1864 the British physicist James Clerk Maxwell made the remarkable suggestion that accelerated electric charges generate linked electric and magnetic disturbances that can travel indefinitely through space. If the charges oscillate periodically, the distur- bances are waves whose electric and magnetic components are perpendicular to each other and to the direction of propagation, as in Fig. 2.1.

From the earlier work of Faraday, Maxwell knew that a changing magnetic field can induce a current in a wire loop. Thus a changing magnetic field is equivalent in its effects to an electric field. Maxwell proposed the converse: a changing electric field has a magnetic field associated with it. The electric fields produced by electromagnetic induction are easy to demonstrate because metals offer little resistance to the flow of charge. Even a weak field can lead to a measurable current in a metal. Weak magnetic fields are much harder to detect, however, and Maxwell’s hypothesis was based on a symmetry argument rather than on experimental findings.

Figure 2.1 The electric and magnetic fields in an electromagnetic wave vary together. The fields are perpendicular to each other and to the direction of propagation of the wave.

Electric field

Direction of wave Magnetic field

54

Chapter Two

If Maxwell was right, electromagnetic (em) waves must occur in which constantly varying electric and magnetic fields are coupled together by both electromagnetic in- duction and the converse mechanism he proposed. Maxwell was able to show that the speed cof electromagnetic waves in free space is given by

c 2.99810⁸m/s

where 0is the electric permittivity of free space and 0is its magnetic permeability.

This is the same as the speed of light waves. The correspondence was too great to be accidental, and Maxwell concluded that light consists of electromagnetic waves.

During Maxwell’s lifetime the notion of em waves remained without direct experi- mental support. Finally, in 1888, the German physicist Heinrich Hertz showed that em waves indeed exist and behave exactly as Maxwell had predicted. Hertz generated the waves by applying an alternating current to an air gap between two metal balls. The width of the gap was such that a spark occurred each time the current reached a peak.

A wire loop with a small gap was the detector; em waves set up oscillations in the loop that produced sparks in the gap. Hertz determined the wavelength and speed of the waves he generated, showed that they have both electric and magnetic components, and found that they could be reflected, refracted, and diffracted.

Light is not the only example of an em wave. Although all such waves have the same fundamental nature, many features of their interaction with matter depend upon

1 00

James Clerk Maxwell (1831–

1879) was born in Scotland shortly before Michael Faraday discovered electromagnetic induc- tion. At nineteen he entered Cam- bridge University to study physics and mathematics. While still a stu- dent, he investigated the physics of color vision and later used his ideas to make the first color pho- tograph. Maxwell became known to the scientific world at twenty-four when he showed that the rings of Saturn could not be solid or liquid but must consist of separate small bodies. At about this time Maxwell became in- terested in electricity and magnetism and grew convinced that the wealth of phenomena Faraday and others had discovered were not isolated effects but had an underlying unity of some kind. Maxwell’s initial step in establishing that unity came in 1856 with the paper “On Faraday’s Lines of Force,” in which he developed a mathematical description of electric and mag- netic fields.

Maxwell left Cambridge in 1856 to teach at a college in Scotland and later at King’s College in London. In this period he expanded his ideas on electricity and magnetism to create a single comprehensive theory of electromagnetism. The funda- mental equations he arrived at remain the foundations of the subject today. From these equations Maxwell predicted that electromagnetic waves should exist that travel with the speed

of light, described the properties the waves should have, and surmised that light consisted of electromagnetic waves. Sadly, he did not live to see his work confirmed in the experiments of the German physicist Heinrich Hertz.

Maxwell’s contributions to kinetic theory and statistical mechanics were on the same profound level as his contribu- tions to electromagnetic theory. His calculations showed that the viscosity of a gas ought to be independent of its pressure, a surprising result that Maxwell, with the help of his wife, con- firmed in the laboratory. They also found that the viscosity was proportional to the absolute temperature of the gas. Maxwell’s explanation for this proportionality gave him a way to estimate the size and mass of molecules, which until then could only be guessed at. Maxwell shares with Boltzmann credit for the equa- tion that gives the distribution of molecular energies in a gas.

In 1865 Maxwell returned to his family’s home in Scotland.

There he continued his research and also composed a treatise on electromagnetism that was to be the standard text on the subject for many decades. It was still in print a century later.

In 1871 Maxwell went back to Cambridge to establish and direct the Cavendish Laboratory, named in honor of the pio- neering physicist Henry Cavendish. Maxwell died of cancer at the age of forty-eight in 1879, the year in which Albert Ein- stein was born. Maxwell had been the greatest theoretical physi- cist of the nineteenth century; Einstein was to be the greatest theoretical physicist of the twentieth century. (By a similar coincidence, Newton was born in the year of Galileo’s death.)

Frequency, Hz 10²² 10²¹ 10²⁰ 10¹⁹ 10¹⁸ 10¹⁷ 10¹⁶ 10¹⁵ 10¹⁴ 10¹³ 10¹² 10¹¹ 10¹⁰ 10⁹ 10⁸ 10⁷ 10⁶ 10⁵ 10⁴ 10³ (1 GHz)

(1 MHz)

(1 kHz)

Photon energy, eV

10⁷ 10⁶ 10⁵ 10⁴ 10³ 10² 10

1 10^–1 10^–2 10^–3 10^–4 10^–5 10^–6 10^–7 10^–8 10^–9 10^–10 10^–11 (1 MeV)

(1 keV)

Radiation

Wavelength, m

10^–13 10^–12 10^–11 10^–10 10^–9 10^–8 10^–7 10^–6 10^–5 10^–4 10^–3 10^–2 10^–1 1 10 10² 10³ 10⁴ 10⁵ (1 pm)

(1 nm)

(1 µm)

(1 mm) (1 cm)

(1 km) Gamma raysUltra- violetInfraredRadio

Visible

TV, FM

Standard broadcast X-raysMicro- waves

Figure 2.2 The spectrum of electromagnetic radiation.

their frequencies. Light waves, which are em waves the eye responds to, span only a brief frequency interval, from about 4.3 10¹⁴Hz for red light to about 7.5 10¹⁴ Hz for violet light. Figure 2.2 shows the em wave spectrum from the low frequencies used in radio communication to the high frequencies found in x-rays and gamma rays.

A characteristic property of all waves is that they obey the principle of superposition:

When two or more waves of the same nature travel past a point at the same time, the instantaneous amplitude there is the sum of the instantaneous amplitudes of the individual waves.

Instantaneous amplitude refers to the value at a certain place and time of the quan- tity whose variations constitute the wave. (“Amplitude” without qualification refers to the maximum value of the wave variable.) Thus the instantaneous amplitude of a wave in a stretched string is the displacement of the string from its normal position; that of a water wave is the height of the water surface relative to its normal level; that of a sound wave is the change in pressure relative to the normal pressure. Since the elec- tric and magnetic fields in a light wave are related by E cB, its instantaneous amplitude can be taken as either E or B. Usually Eis used, since it is the electric fields of light waves whose interactions with matter give rise to nearly all common optical effects.

A C

B D

The interference of water waves. Constructive interference occurs along the line ABand destructive interference occurs along the line CD.

+

(a) b)

+ = =

(

Figure 2.3(a) In constructive interference, superposed waves in phase reinforce each other. (b) In destructive interference, waves out of phase partially or completely cancel each other.

When two or more trains of light waves meet in a region, they interfereto produce a new wave there whose instantaneous amplitude is the sum of those of the original waves. Constructive interference refers to the reinforcement of waves with the same phase to produce a greater amplitude, and destructive interference refers to the partial or complete cancellation of waves whose phases differ (Fig. 2.3). If the original waves have different frequencies, the result will be a mixture of constructive and destructive interference, as in Fig. 3.4.

The interference of light waves was first demonstrated in 1801 by Thomas Young, who used a pair of slits illuminated by monochromatic light from a single source (Fig. 2.4).

From each slit secondary waves spread out as though originating at the slit; this is an ex- ample of diffraction,which, like interference, is a characteristic wave phenomenon. Ow- ing to interference, the screen is not evenly lit but shows a pattern of alternate bright and dark lines. At those places on the screen where the path lengths from the two slits differ by an odd number of half wavelengths (2, 32, 52, . . .), destructive inter- ference occurs and a dark line is the result. At those places where the path lengths are

56

Chapter Two

Constructive interference produces bright line Destructive interference produces dark line Constructive interference produces bright line Monochromatic

light source

Appearance of screen Figure 2.4 Origin of the interference pattern in Young’s experiment. Constructive interference occurs where the difference in path lengths from the slits to the screen is , , 2, . . . . Destructive interference occurs where the path difference is 2, 32, 52, . . . .

equal or differ by a whole number of wavelengths (, 2, 3, . . .), constructive inter- ference occurs and a bright line is the result. At intermediate places the interference is only partial, so the light intensity on the screen varies gradually between the bright and dark lines.

Interference and diffraction are found only in waves—the particles we are familiar with do not behave in those ways. If light consisted of a stream of classical particles, the entire screen would be dark. Thus Young’s experiment is proof that light consists of waves. Maxwell’s theory further tells us what kind of waves they are: electromag- netic. Until the end of the nineteenth century the nature of light seemed settled forever.

2.2 BLACKBODY RADIATION

Only the quantum theory of light can explain its origin

Following Hertz’s experiments, the question of the fundamental nature of light seemed clear: light consisted of em waves that obeyed Maxwell’s theory. This cer- tainty lasted only a dozen years. The first sign that something was seriously amiss came from attempts to understand the origin of the radiation emitted by bodies of matter.

We are all familiar with the glow of a hot piece of metal, which gives off visible light whose color varies with the temperature of the metal, going from red to yellow to white as it becomes hotter and hotter. In fact, other frequencies to which our eyes do not respond are present as well. An object need not be so hot that it is luminous for it to be radiating em energy; all objects radiate such energy continuously whatever their temperatures, though which frequencies predominate depends on the temperature. At room temperature most of the radiation is in the infrared part of the spectrum and hence is invisible.

The ability of a body to radiate is closely related to its ability to absorb radiation.

This is to be expected, since a body at a constant temperature is in thermal equilib- rium with its surroundings and must absorb energy from them at the same rate as it emits energy. It is convenient to consider as an ideal body one that absorbs allradi- ation incident upon it, regardless of frequency. Such a body is called a blackbody.

The point of introducing the idealized blackbody in a discussion of thermal ra- diation is that we can now disregard the precise nature of whatever is radiating, since

2 ✕ 10¹⁴ T = 1800 K

0 4 ✕ 10¹⁴ 6 ✕ 10¹⁴ Hz Visible light Frequency, v

Spectral energy density, u(v)dv

T = 1200 K

Figure 2.6 Blackbody spectra. The spectral distribution of energy in the radiation depends only on the temperature of the body. The higher the temperature, the greater the amount of radiation and the higher the frequency at which the maximum emission occurs. The dependence of the latter frequency on temperature follows a formula called Wien’s displacement law, which is discussed in Sec. 9.6.

Incident Light ray

Figure 2.5 A hole in the wall of a hollow object is an excellent ap- proximation of a blackbody.

The color and brightness of an object heated until it glows, such as the filament of this light bulb, depends upon its temperature, which here is about 3000 K. An object that glows white is hotter than it is when it glows red, and it gives off more light as well.

all blackbodies behave identically. In the laboratory a blackbody can be approximated by a hollow object with a very small hole leading to its interior (Fig. 2.5). Any ra- diation striking the hole enters the cavity, where it is trapped by reflection back and forth until it is absorbed. The cavity walls are constantly emitting and absorbing ra- diation, and it is in the properties of this radiation (blackbody radiation) that we are interested.

Experimentally we can sample blackbody radiation simply by inspecting what emerges from the hole in the cavity. The results agree with everyday experience. A blackbody radiates more when it is hot than when it is cold, and the spectrum of a hot blackbody has its peak at a higher frequency than the peak in the spectrum of a cooler one. We recall the behavior of an iron bar as it is heated to progressively higher temperatures: at first it glows dull red, then bright orange-red, and eventually it be- comes “white hot.” The spectrum of blackbody radiation is shown in Fig. 2.6 for two temperatures.

The Ultraviolet Catastrophe

Why does the blackbody spectrum have the shape shown in Fig. 2.6? This prob- lem was examined at the end of the nineteenth century by Lord Rayleigh and James Jeans. The details of their calculation are given in Chap. 9. They started by con- sidering the radiation inside a cavity of absolute temperature T whose walls are perfect reflectors to be a series of standing em waves (Fig. 2.7). This is a three- dimensional generalization of standing waves in a stretched string. The condition

58

Chapter Two

λ = 2L

L

λ = L λ =2L

3

Figure 2.7 Em radiation in a cav- ity whose walls are perfect reflec- tors consists of standing waves that have nodes at the walls, which restricts their possible wavelengths. Shown are three possible wavelengths when the distance between opposite walls is L.

for standing waves in such a cavity is that the path length from wall to wall, whatever the direction, must be a whole number of half-wavelengths, so that a node occurs at each reflecting surface. The number of independent standing waves G()d in the frequency interval between and dper unit volume in the cavity turned out to be

G()d (2.1)

This formula is independent of the shape of the cavity. As we would expect, the higher the frequency ,the shorter the wavelength and the greater the number of possible standing waves.

The next step is to find the average energy per standing wave. According to the theorem of equipartition of energy,a mainstay of classical physics, the average energy per degree of freedom of an entity (such as a molecule of an ideal gas) that is a mem- ber of a system of such entities in thermal equilibrium at the temperature T is ¹2kT.

Here kis Boltzmann’s constant:

Boltzmann’s constant k1.38110²³J/K

A degree of freedom is a mode of energy possession. Thus a monatomic ideal gas molecule has three degrees of freedom, corresponding to kinetic energy of motion in three independent directions, for an average total energy of ³2kT.

A one-dimensional harmonic oscillator has two degrees of freedom, one that corre- sponds to its kinetic energy and one that corresponds to its potential energy. Because each standing wave in a cavity originates in an oscillating electric charge in the cavity wall, two degrees of freedom are associated with the wave and it should have an average energy of 2(¹2)kT:

kT (2.2)

The total energy u() dper unit volume in the cavity in the frequency interval from to dis therefore

u() dG() d ²d (2.3)

This radiation rate is proportional to this energy density for frequencies between and d. Equation (2.3), the Rayleigh-Jeans formula, contains everything that classi- cal physics can say about the spectrum of blackbody radiation.

Even a glance at Eq. (2.3) shows that it cannot possibly be correct. As the fre- quency increases toward the ultraviolet end of the spectrum, this formula predicts that the energy density should increase as ². In the limit of infinitely high fre- quencies, u() dtherefore should also go to infinity. In reality, of course, the energy density (and radiation rate) falls to 0 as S(Fig. 2.8). This discrepancy became known as the ultraviolet catastropheof classical physics. Where did Rayleigh and Jeans go wrong?

8kT c³ Rayleigh-Jeans

formula

Classical average energy per standing wave

8²d c³ Density of standing

waves in cavity

60

Chapter Two

Planck Radiation Formula

In 1900 the German physicist Max Planck used “lucky guesswork” (as he later called it) to come up with a formula for the spectral energy density of blackbody radiation:

u() d (2.4)

Here his a constant whose value is

Planck’s constant h6.62610³⁴Js ³d e^hkT1 8h

c³ Planck radiation

formula

0

Spectral energy density, u(v)dv

1 ✕ 10¹⁴ 2 ✕ 10¹⁴ 3 ✕ 10¹⁴ 4 ✕ 10¹⁴ Frequency, v (Hz)

Rayleigh-Jeans

Observed

Figure 2.8 Comparison of the Rayleigh-Jeans formula for the spectrum of the radiation from a black- body at 1500 K with the observed spectrum. The discrepancy is known as the ultraviolet catastrophe because it increases with increasing frequency. This failure of classical physics led Planck to the dis- covery that radiation is emitted in quanta whose energy is h.

Max Planck (1858–1947) was born in Kiel and educated in Mu- nich and Berlin. At the University of Berlin he studied under Kirch- hoff and Helmholtz, as Hertz had done earlier. Planck realized that blackbody radiation was important because it was a fundamental effect independent of atomic structure, which was still a mystery in the late nineteenth century, and worked at understanding it for six years be- fore finding the formula the radiation obeyed. He “strived from the day of its discovery to give it a real physical interpretation.”

The result was the discovery that radiation is emitted in energy steps of h. Although this discovery, for which he received the Nobel Prize in 1918, is now considered to mark the start of

modern physics, Planck himself remained skeptical for a long time of the physical reality of quanta. As he later wrote, “My vain attempts to somehow reconcile the elementary quantum with classical theory continued for many years and cost me great effort. . . . Now I know for certain that the quantum of action has a much more fundamental significance than I orig- inally suspected.”

Like many physicists, Planck was a competent musician (he sometimes played with Einstein) and in addition enjoyed moun- tain climbing. Although Planck remained in Germany during the Hitler era, he protested the Nazi treatment of Jewish scien- tists and lost his presidency of the Kaiser Wilhelm Institute as a result. In 1945 one of his sons was implicated in a plot to kill Hitler and was executed. After World War II the Institute was renamed after Planck and he was again its head until his death.